Which is the smallest integer that can be expressed as a sum of consecutive integers in a given number of ways?

Alternating iterations of the Smarandache function and the Euler phi-function respectively the sum of divisors function. Some light is thrown on loops and invariants resulting from these iterations. An important question is resolved with the amazing involvement of the famous Fermat numbers.

A particularly interesting subject is the Smarandache partial perfect additive sequence, it has a simple definition and a strange behaviour.

Smarandache general continued fractions are treated in great detail and proof is given for the convergence under specified conditions.

Smarandache k-k additive relationships as well as subtractive relationships are treated with some observations on the occurrence of prime twins.

A substantial part is devoted to concatenation and deconcatenation problems. Some divisibilty properties of very large numbers is studied. In particular some questions raised on the Smarandache deconstructive sequence are resolved.